Certain operators and operator algebras in perturbation theory and on function spaces
Jingbo Xia Principal Investigator
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The proposer will continue the research that he has<br/>been pursuing under the support of the National Science Foundation. The<br/>specific areas of the proposed research include the following: (1)<br/>Simultaneous diagonalization of commuting tuples of self-adjoint operators<br/>modulo various norm ideals. This problem has been solved (with NSF<br/>support) in the case where the norm ideal is the Schatten p-class when p <br/>is strictly greater than 1. The proposer will next consider the case<br/>where p is 1, i.e., where the norm ideal is the trace class. This is<br/>a difficult problem, but this is also an important problem because of its<br/>potential applications. The proposer will also consider a class of<br/>ideals which are related to the Schatten class. (2) The complete<br/>determination of automorphisms of the full Toeplitz algebra on the unit<br/>circle which are induced by homeomorphisms of the circle. This has been<br/>accomplished under previous NSF support in the case where the<br/>homeomorphism in question is bi-Lipschitz. The final goal is to remove<br/>the bi-Lipschitz condition. This involves some careful estimates of norms<br/>in the Toeplitz algebra and the use of certain singular integral operators. <br/>(3) Toeplitz algebras associated with minimal flows. The ultimate goal<br/>here is to use K-theory to characterize the invertibility of systems of <br/>Toeplitz operators associated with such flows. (4) Hankel operators on<br/>certain reproducing-kernel Hilbert spaces. Here the main question is the<br/>Schatten-class membership of these operator. The reproducing kernel<br/>will be involved in certain quantitative estimates.<br/><br/>The proposed problems are fairly representative of the current<br/>research interests in operator theory and operator algebras, which is a<br/>study of, among other things, the spectral properties of various linear<br/>operators. In part inspired and demanded by the development of the<br/>quantum theory in the early part of the 20th century, this study was<br/>initiated by great mathematicians such as H. Weyl and J. von Neumann.<br/>Because additivity (i.e., linearity) appears in many fundamental aspects <br/>of nature, operator theory provides the right mathematical tools for<br/>scientific fields ranging from atomic physics to optimal control. Many<br/>abstract problems in operator theory and operator algebras owe their<br/>origin to these fields of applications. For example, both for theoretical<br/>reasons and for practical applications, quite often one must deal with, <br/>or introduce, perturbations which are "small" by some measure or other.<br/>Problem (1) is about such perturbations. The root of this problem can be<br/>traced back to a paper of Weyl published in 1909, which asserts that a<br/>continuous spectrum can be turned into a discrete one by a compact <br/>(which a measure of "smallness") perturbation. Problem (2) requires both<br/>modern techniques and classical-style mathematical analysis. A theme<br/>which underlies all these problems is the establishment of various <br/>estimates (i.e., bounds or growth rates). In general, the sharper the<br/>estimates, the better theorems one obtains.